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Open Access

Abstract

Let MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function
space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ)
is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂
L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.